Remarks on nonlinear equations with measures

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July 2013, 12(4): 1745-1753. doi: 10.3934/cpaa.2013.12.1745

Remarks on nonlinear equations with measures

1.	Department of Mathematics, Technion, Haifa 32000, Israel

Received February 2011 Revised July 2012 Published November 2012

Abstract
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We study the Dirichlet boundary value problem for equations with absorption of the form $-\Delta u+g\circ u=\mu$ in a bounded domain $\Omega\subset R^N$ where $g$ is a continuous odd monotone increasing function. Under some additional assumptions on $g$, we present necessary and sufficient conditions for existence when $\mu$ is a finite measure. We also discuss the notion of solution when the measure $\mu$ is positive and blows up on a compact subset of $\Omega$.

Keywords: Orlicz - Sobolev spaces, absorption term, Semi-linear elliptic equation, weak solution, capacities..

Mathematics Subject Classification: Primary: 35J65, 46E30; Secondary: 46E35, 32U2.

Citation: Moshe Marcus. Remarks on nonlinear equations with measures. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1745-1753. doi: 10.3934/cpaa.2013.12.1745

References:

[1]	D. R. Adams and L. I. Hedberg, "Function Spaces and Potential Theory,'', Grundlehren Math. Wissen., 314 (1996).
[2]	N. Aissaoui and A. Benkirane, Capacité dans les espaces d'Orlicz,, Ann. Sci. Math. Qu\'ebec, 12 (1994), 1.
[3]	P. Baras and M. Pierre, Singularitès éliminables pour des équations semilinèaires,, Ann. Inst. Fourier, 34 (1984), 185.
[4]	D. Bartolucci, F. Leoni, L. Orsina and A. Ponce, Semilinear equations with exponential nonlinearity and measure data,, Ann. I. H. Poincar\'e - AN, 22 (2005), 799.
[5]	Ph. Benilan and H. Brezis, Nonlinear preoblems related to the Thomas-Fermi equation,, J. Evolution Eq., 3 (2003), 673.
[6]	H. Brezis, Notes, unpublished (circa 1970)., (1970).
[7]	H. Brezis H. and W. Strauss, Semilinear second-order elliptic equations in $L^1$,, J. Math. Soc. Japan, 25 (1973), 565.
[8]	Th. K. Donaldson and N. S. Trudinger, Orlicz-Sobolev spaces and imbedding theorems,, J. Funct. An., 8 (1971), 52.
[9]	M. A. Krasnoselskkii and Y. B. Rutickii, "Convex Functions and Orlicz Spaces,'', P. Noordhoff, (1961).
[10]	J. B. Keller, On solutions of $\Delta u = f(u)$,, Comm. Pure Appl. Math., 10 (1957), 503.
[11]	M. Marcus and L. Véron, The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case,, Arch. Rat. Mech. Anal., 144 (1998), 201.
[12]	M. Marcus and L. Véron, Capacitary estimates of positive solutions of semilinear elliptic equations with absorption,, J. European Math. Soc., 6 (2004), 483.
[13]	R. Osserman, On the inequality $\Delta u\geq f(u)$,, Pacific J. Math., 7 (1957), 1641.
[14]	L. Véron, "Singularities of Solutions of Second Order Quasilinear Equations,'', Pitman Research Notes, 353 (1996).

show all references

References:

[1]	D. R. Adams and L. I. Hedberg, "Function Spaces and Potential Theory,'', Grundlehren Math. Wissen., 314 (1996).
[2]	N. Aissaoui and A. Benkirane, Capacité dans les espaces d'Orlicz,, Ann. Sci. Math. Qu\'ebec, 12 (1994), 1.
[3]	P. Baras and M. Pierre, Singularitès éliminables pour des équations semilinèaires,, Ann. Inst. Fourier, 34 (1984), 185.
[4]	D. Bartolucci, F. Leoni, L. Orsina and A. Ponce, Semilinear equations with exponential nonlinearity and measure data,, Ann. I. H. Poincar\'e - AN, 22 (2005), 799.
[5]	Ph. Benilan and H. Brezis, Nonlinear preoblems related to the Thomas-Fermi equation,, J. Evolution Eq., 3 (2003), 673.
[6]	H. Brezis, Notes, unpublished (circa 1970)., (1970).
[7]	H. Brezis H. and W. Strauss, Semilinear second-order elliptic equations in $L^1$,, J. Math. Soc. Japan, 25 (1973), 565.
[8]	Th. K. Donaldson and N. S. Trudinger, Orlicz-Sobolev spaces and imbedding theorems,, J. Funct. An., 8 (1971), 52.
[9]	M. A. Krasnoselskkii and Y. B. Rutickii, "Convex Functions and Orlicz Spaces,'', P. Noordhoff, (1961).
[10]	J. B. Keller, On solutions of $\Delta u = f(u)$,, Comm. Pure Appl. Math., 10 (1957), 503.
[11]	M. Marcus and L. Véron, The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case,, Arch. Rat. Mech. Anal., 144 (1998), 201.
[12]	M. Marcus and L. Véron, Capacitary estimates of positive solutions of semilinear elliptic equations with absorption,, J. European Math. Soc., 6 (2004), 483.
[13]	R. Osserman, On the inequality $\Delta u\geq f(u)$,, Pacific J. Math., 7 (1957), 1641.
[14]	L. Véron, "Singularities of Solutions of Second Order Quasilinear Equations,'', Pitman Research Notes, 353 (1996).

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